We study wrappings of the unit sphere by a piece of paper
(or, perhaps more accurately, a piece of foil).
Such wrappings differ from standard origami because they require
infinitely many infinitesimally small “folds” in order to transform
the flat sheet into a positive-curvature sphere.
Our goal is to find shapes that have small area even when expanded to
shapes that tile the plane.
We characterize the smallest square that wraps the unit sphere,
show that a 0.1% smaller equilateral triangle suffices,
and find a 20% smaller shape that still tiles the plane.